Select The Correct Answer.A Circle Is Described By The Equation $x^2 + Y^2 - 6x + 8y = 0$. What Are The Coordinates Of The Center Of The Circle And The Length Of Its Radius?A. $(3, -4), 25$ Units B. $(-3, 4), 25$ Units C.

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle can be written in the form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. In this article, we will explore how to find the coordinates of the center and the length of the radius of a circle given its equation.

The Equation of a Circle

The equation of a circle is given by $x^2 + y^2 - 6x + 8y = 0$. To find the center and radius of the circle, we need to rewrite the equation in the standard form $(x - h)^2 + (y - k)^2 = r^2$.

Completing the Square

To complete the square, we need to group the $x$ terms and $y$ terms separately. We can do this by rearranging the equation as follows:

$x^2 - 6x + y^2 + 8y = 0$

Next, we need to add and subtract the square of half the coefficient of $x$ and $y$ to complete the square. For $x$, we have:

$(-6/2)^2 = 9$

For $y$, we have:

$(8/2)^2 = 16$

Now, we can add and subtract these values to the equation:

$x^2 - 6x + 9 + y^2 + 8y + 16 = 9 + 16$

Simplifying the equation, we get:

$(x - 3)^2 + (y + 4)^2 = 25$

Finding the Center and Radius

Now that we have the equation in the standard form, we can easily identify the center and radius of the circle. The center of the circle is given by $(h, k) = (3, -4)$, and the radius is given by $r = \sqrt{25} = 5$ units.

Conclusion

In this article, we have shown how to find the coordinates of the center and the length of the radius of a circle given its equation. We used the method of completing the square to rewrite the equation in the standard form and then identified the center and radius of the circle. The center of the circle is given by $(3, -4)$, and the radius is given by $5$ units.

Answer

The correct answer is A. $(3, -4), 5$ units.

Discussion

This problem is a classic example of how to find the center and radius of a circle given its equation. The method of completing the square is a powerful tool in mathematics that can be used to solve a wide range of problems. In this case, we used the method to rewrite the equation of the circle in the standard form and then identified the center and radius of the circle.

Related Topics

• Equation of a Circle: The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.
• Completing the Square: Completing the square is a method used to rewrite a quadratic equation in the standard form.
• Center and Radius of a Circle: The center of a circle is given by $(h, k)$, and the radius is given by $r$.

References

• Mathematics Handbook: A comprehensive guide to mathematics that covers a wide range of topics, including algebra, geometry, and calculus.
• Geometry: A branch of mathematics that deals with the study of shapes and sizes of objects.
• Algebra: A branch of mathematics that deals with the study of variables and their relationships.
  Q&A: Finding the Center and Radius of a Circle =====================================================

Introduction

In our previous article, we showed how to find the coordinates of the center and the length of the radius of a circle given its equation. In this article, we will answer some frequently asked questions related to finding the center and radius of a circle.

Q: What is the equation of a circle?

A: The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.

Q: How do I find the center and radius of a circle?

A: To find the center and radius of a circle, you need to rewrite the equation of the circle in the standard form $(x - h)^2 + (y - k)^2 = r^2$. You can do this by completing the square.

Q: What is completing the square?

A: Completing the square is a method used to rewrite a quadratic equation in the standard form. It involves adding and subtracting the square of half the coefficient of $x$ and $y$ to complete the square.

Q: How do I complete the square?

A: To complete the square, you need to group the $x$ terms and $y$ terms separately. Then, you need to add and subtract the square of half the coefficient of $x$ and $y$ to complete the square.

Q: What is the center of a circle?

A: The center of a circle is given by $(h, k)$, where $(h, k)$ is the point that is equidistant from all points on the circle.

Q: What is the radius of a circle?

A: The radius of a circle is given by $r$, where $r$ is the distance from the center of the circle to any point on the circle.

Q: How do I find the distance between two points?

A: To find the distance between two points, you can use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is given by $A = \pi r^2$, where $r$ is the radius of the circle.

Q: What is the formula for the circumference of a circle?

A: The formula for the circumference of a circle is given by $C = 2\pi r$, where $r$ is the radius of the circle.

Conclusion

In this article, we have answered some frequently asked questions related to finding the center and radius of a circle. We have also provided some formulas and equations that are useful in finding the center and radius of a circle.

Related Topics

• Equation of a Circle: The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.
• Completing the Square: Completing the square is a method used to rewrite a quadratic equation in the standard form.
• Center and Radius of a Circle: The center of a circle is given by $(h, k)$, and the radius is given by $r$.

References

• Mathematics Handbook: A comprehensive guide to mathematics that covers a wide range of topics, including algebra, geometry, and calculus.
• Geometry: A branch of mathematics that deals with the study of shapes and sizes of objects.
• Algebra: A branch of mathematics that deals with the study of variables and their relationships.